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<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.sf_gamma.gamma_ratios"></a><a class="link" href="gamma_ratios.html" title="Ratios of Gamma Functions">Ratios of Gamma Functions</a>
</h3></div></div></div>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">gamma</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
</pre>
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_delta_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">delta</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_delta_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">delta</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>

<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<h5>
<a name="math_toolkit.sf_gamma.gamma_ratios.h0"></a>
        <span class="phrase"><a name="math_toolkit.sf_gamma.gamma_ratios.description"></a></span><a class="link" href="gamma_ratios.html#math_toolkit.sf_gamma.gamma_ratios.description">Description</a>
      </h5>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
        Returns the ratio of gamma functions:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/gamma_ratio0.svg"></span>

        </p></blockquote></div>
<p>
        The final <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
        be used to control the behaviour of the function: how it handles errors,
        what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">policy
        documentation for more details</a>.
      </p>
<p>
        Internally this just calls <code class="computeroutput"><span class="identifier">tgamma_delta_ratio</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span>
        <span class="identifier">b</span><span class="special">-</span><span class="identifier">a</span><span class="special">)</span></code>.
      </p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_delta_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">delta</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_delta_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">delta</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
        Returns the ratio of gamma functions:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/gamma_ratio1.svg"></span>

        </p></blockquote></div>
<p>
        The final <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
        be used to control the behaviour of the function: how it handles errors,
        what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">policy
        documentation for more details</a>.
      </p>
<p>
        Note that the result is calculated accurately even when <span class="emphasis"><em>delta</em></span>
        is small compared to <span class="emphasis"><em>a</em></span>: indeed even if <span class="emphasis"><em>a+delta
        ~ a</em></span>. The function is typically used when <span class="emphasis"><em>a</em></span>
        is large and <span class="emphasis"><em>delta</em></span> is very small.
      </p>
<p>
        The return type of these functions is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
        type calculation rules</em></span></a> when T1 and T2 are different types,
        otherwise the result type is simple T1.
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../graphs/tgamma_delta_ratio.svg" align="middle"></span>

        </p></blockquote></div>
<h5>
<a name="math_toolkit.sf_gamma.gamma_ratios.h1"></a>
        <span class="phrase"><a name="math_toolkit.sf_gamma.gamma_ratios.accuracy"></a></span><a class="link" href="gamma_ratios.html#math_toolkit.sf_gamma.gamma_ratios.accuracy">Accuracy</a>
      </h5>
<p>
        The following table shows the peak errors (in units of epsilon) found on
        various platforms with various floating point types. Unless otherwise specified
        any floating point type that is narrower than the one shown will have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively zero error</a>.
      </p>
<div class="table">
<a name="math_toolkit.sf_gamma.gamma_ratios.table_tgamma_delta_ratio"></a><p class="title"><b>Table 8.7. Error rates for tgamma_delta_ratio</b></p>
<div class="table-contents"><table class="table" summary="Error rates for tgamma_delta_ratio">
<colgroup>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
              </th>
<th>
                <p>
                  GNU C++ version 7.1.0<br> linux<br> double
                </p>
              </th>
<th>
                <p>
                  GNU C++ version 7.1.0<br> linux<br> long double
                </p>
              </th>
<th>
                <p>
                  Sun compiler version 0x5150<br> Sun Solaris<br> long double
                </p>
              </th>
<th>
                <p>
                  Microsoft Visual C++ version 14.1<br> Win32<br> double
                </p>
              </th>
</tr></thead>
<tbody>
<tr>
<td>
                <p>
                  tgamma + small delta ratios
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0ε (Mean = 0ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 5.83ε (Mean = 1.3ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 15.4ε (Mean = 2.09ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 7.56ε (Mean = 1.31ε)</span>
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  tgamma + small delta ratios (negative delta)
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0ε (Mean = 0ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 7.94ε (Mean = 1.4ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 18.3ε (Mean = 2.03ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 7.43ε (Mean = 1.42ε)</span>
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  tgamma + small integer ratios
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0ε (Mean = 0ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 1.96ε (Mean = 0.677ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 1.96ε (Mean = 0.677ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 2.74ε (Mean = 0.736ε)</span>
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  tgamma + small integer ratios (negative delta)
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0ε (Mean = 0ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 1.62ε (Mean = 0.451ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 1.62ε (Mean = 0.451ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 2.15ε (Mean = 0.685ε)</span>
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  integer tgamma ratios
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0ε (Mean = 0ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0.997ε (Mean = 0.4ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0.997ε (Mean = 0.4ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0.968ε (Mean = 0.386ε)</span>
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  integer tgamma ratios (negative delta)
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0ε (Mean = 0ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0.853ε (Mean = 0.176ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0.853ε (Mean = 0.176ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0.974ε (Mean = 0.175ε)</span>
                </p>
              </td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.sf_gamma.gamma_ratios.table_tgamma_ratio"></a><p class="title"><b>Table 8.8. Error rates for tgamma_ratio</b></p>
<div class="table-contents"><table class="table" summary="Error rates for tgamma_ratio">
<colgroup>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
              </th>
<th>
                <p>
                  GNU C++ version 7.1.0<br> linux<br> double
                </p>
              </th>
<th>
                <p>
                  GNU C++ version 7.1.0<br> linux<br> long double
                </p>
              </th>
<th>
                <p>
                  Sun compiler version 0x5150<br> Sun Solaris<br> long double
                </p>
              </th>
<th>
                <p>
                  Microsoft Visual C++ version 14.1<br> Win32<br> double
                </p>
              </th>
</tr></thead>
<tbody><tr>
<td>
                <p>
                  tgamma ratios
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0.694ε (Mean = 0.0347ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 2.99ε (Mean = 1.15ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 174ε (Mean = 61.2ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 3.28ε (Mean = 1.12ε)</span>
                </p>
              </td>
</tr></tbody>
</table></div>
</div>
<br class="table-break"><h5>
<a name="math_toolkit.sf_gamma.gamma_ratios.h2"></a>
        <span class="phrase"><a name="math_toolkit.sf_gamma.gamma_ratios.testing"></a></span><a class="link" href="gamma_ratios.html#math_toolkit.sf_gamma.gamma_ratios.testing">Testing</a>
      </h5>
<p>
        Accuracy tests use data generated at very high precision (with <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL
        RR class</a> set at 1000-bit precision: about 300 decimal digits) and
        a deliberately naive calculation of Γ(x)/Γ(y).
      </p>
<h5>
<a name="math_toolkit.sf_gamma.gamma_ratios.h3"></a>
        <span class="phrase"><a name="math_toolkit.sf_gamma.gamma_ratios.implementation"></a></span><a class="link" href="gamma_ratios.html#math_toolkit.sf_gamma.gamma_ratios.implementation">Implementation</a>
      </h5>
<p>
        The implementation of these functions is very similar to that of <a class="link" href="../sf_beta/beta_function.html" title="Beta">beta</a>,
        and is based on combining similar power terms to improve accuracy and avoid
        spurious overflow/underflow.
      </p>
<p>
        In addition there are optimisations for the situation where <span class="emphasis"><em>delta</em></span>
        is a small integer: in which case this function is basically the reciprocal
        of a rising factorial, or where both arguments are smallish integers: in
        which case table lookup of factorials can be used to calculate the ratio.
      </p>
</div>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
      Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
      Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
      Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
      </p>
</div>
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